Can new beings come into existence without this existence being a rebirth?

In the Pali suttas, the word "a being" ("satta") refers to "a view" generated from "strong attachment", as follows:

Why now do you assume ‘a being’? Mara, is that your speculative view? This is a heap of sheer formations: Here no being is found. Just as, with an assemblage of parts, The word ‘chariot’ is used, So, when the aggregates exist, There is the convention ‘a being.’

SN 5.10

'A being,' lord. 'A being,' it's said. To what extent is one said to be 'a being'?

Any desire, passion, delight, or craving for form, Radha: when one is caught up there, tied up there, one is said to be 'a being.'

Any desire, passion, delight, or craving for feeling... perception... fabrications...

Any desire, passion, delight, or craving for consciousness, Radha: when one is caught up there, tied up there, one is said to be 'a being.'

Just as when boys or girls are playing with little sand castles: as long as they are not free from passion, desire, love, thirst, fever, & craving for those little sand castles, that's how long they have fun with those sand castles, enjoy them, treasure them, feel possessive of them.

SN 23.2

MN 64 says a new born child has no "self-identity" therefore has no notion of "a being", as follows:

For a young tender infant lying prone does not even have the notion ‘identity,’ so how could identity view arise in him?

A young tender infant lying prone does not even have the notion ‘beings,’ so how could ill will towards beings arise in him?

MN 38 clearly explains it is only when a child reaches a certain age does the idea of "a being" or "a birth" arises in their mind, as follows:

The mother then carries the embryo in her womb for nine or ten months with much anxiety, as a heavy burden. Then, at the end of nine or ten months, the mother gives birth with much anxiety, as a heavy burden. Then, when the child is born, she nourishes it with her own blood; for the mother’s breast-milk is called blood in the Noble One’s Discipline.

When he grows up and his faculties mature, the child plays at such games as toy ploughs, tipcat, somersaults, toy windmills, toy measures, toy carts, and a toy bow and arrow.

When he grows up and his faculties mature still further, the youth enjoys himself provided and endowed with the five cords of sensual pleasure, with forms cognizable by the eye… sounds cognizable by the ear…odours cognizable by the nose… flavours cognizable by the tongue…tangibles cognizable by the body that are wished for, desired, agreeable and likeable, connected with sensual desire, and provocative of lust.

On seeing a form with the eye, he lusts after it if it is pleasing; he dislikes it if it is unpleasing. He abides with mindfulness of the body unestablished, with a limited mind, and he does not understand as it actually is the deliverance of mind and deliverance by wisdom wherein those evil unwholesome states cease without remainder. Engaged as he is in favouring and opposing, whatever feeling he feels—whether pleasant or painful or neither-painful-nor-pleasant—he delights in that feeling, welcomes it, and remains holding to it. As he does so, delight arises in him. Now delight in feelings is clinging. With his clinging as condition, being comes to be; with being as condition, birth; with birth as condition, ageing and death, sorrow, lamentation, pain, grief, and despair come to be. Such is the origin of this whole mass of suffering.

On hearing a sound with the ear… On smelling an odour with the nose… On tasting a flavour with the tongue… On touching a tangible with the body… On cognizing a mind-object with the mind, he lusts after it if it is pleasing; he dislikes it if it is unpleasing…Now delight in feelings is clinging. With his clinging as condition, being comes to be; with being as condition, birth; with birth as condition, ageing and death, sorrow, lamentation, pain, grief, and despair come to be. Such is the origin of this whole mass of suffering.

It may be noted the eternalist translator of the above titles the above section as "The Continuation of the Round". But his title is obviously his own addition to the text & wrong. It should be titled "The Start/Commencement of the Round". Therefore, there is a time when a child has its first "birth"; its first mental conception of "a being" and "beings".

Also, it should be noted the word "birth" ("jati") means the birth of the view or conception of "beings", as follows:

And what is birth? Whatever birth, taking birth, descent, coming-to-be, coming-forth, appearance [manifestation] of aggregates & acquisition of [sense] media of the various beings in this or that group of beings, that is called birth.

SN 12.2

It should also be noted many will vomit up hot blood when their 'being' reads this answer.

Ruben's answer does a good job of addressing what appears to be your primary question, namely what does Buddhism have to say about the beginning of a being's existence? I would like to address, perhaps more explicitly, the secondary question posed in your third paragraph, namely "(are there) a set number of souls or spirits that just recycle through rebirth(?)".

Within the Pali Canon, a hint towards the Buddha's teaching on this matter can be found in the 46th sutta of the 12th samyutta of the samyutta nikaya (conventionaly denoted SN 12.46).

At Savatthī. Then a certain brahmin approached the Blessed One and exchanged greetings with him. When they had concluded their greetings and cordial talk, he sat down to one side and said to him:

“How is it, Master Gotama: is the one who acts the same as the one who experiences the result?”

“‘The one who acts is the same as the one who experiences the result': this, brahmin, is one extreme.”

“Then, Master Gotama, is the one who acts one, and the one who experiences the result another?”

“‘The one who acts is one, and the one who experiences the result is another’: this, brahmin, is the second extreme. Without veering towards either of these extremes, the Tathagata teaches the Dhamma by the middle: ‘With ignorance as condition, volitional formations come to be; with volitional formations as condition, consciousness…. Such is the origin of this whole mass of suffering. But with the remainderless fading away and cessation of ignorance comes cessation of volitional formations; with the cessation of volitional formations, cessation of consciousness…. Such is the cessation of this whole mass of suffering.’”

When this was said, that brahmin said to the Blessed One: “Magnificent, Master Gotama!… I go for refuge to Master Gotama, and to the Dhamma, and to the Bhikkhu Saṅgha. From today let Master Gotama remember me as a lay follower who has gone for refuge for life.” (Translation by Bhikkhu Bodhi)

I think this sutta dispels the idea of a "one-to-one mapping" between lives, which your question seems to imply. A rather stronger statement on the matter of "a set number of souls" can be found in Mahayana sources. Consider, for example, the following passage from the "Diamond Sutra", which is framed as a conversation between the monk Subhuti and the Buddha (Tathagata):

“Subhuti, do not say that the Tathagata has the idea, ‘I will bring living beings to the shore of liberation.’ Do not think that way, Subhuti. Why? In truth there is no living being for the Tathagata to bring to the other shore. If the Tathagata were to think there was, he would be caught in the idea of a self, a person, a living being, or a life span." (Translated by Thich Naht Hahn)

Both of these suttas illustrate two ideas idea that you will encounter again and again in Buddhist teachings. My own crude summary is: the correct way to understand the World is in terms of processes, rather than objects (cf. substance vs bundle theory in Western Philosophy); concern yourself with the cessation of suffering, not the construction of "complete" metaphysical theories.

The beginnings of samsara and the precise working out of kamma is basically unknown, unknowable and may bring madness and vexation to anyone speculating on them.

Transmigration (samsara) has no known beginning. No first point is found of sentient beings roaming and transmigrating, hindered by ignorance and fettered by craving. For such a long time you have undergone suffering, agony, and disaster, swelling the cemeteries. This is quite enough for you to become disillusioned, dispassionate, and freed regarding all conditions.
SN 15.3 (translated by Ven. Sujato)

From an inconstruable beginning comes transmigration (samsara). A beginning point is not evident, though beings hindered by ignorance and fettered by craving are transmigrating & wandering on. Long have you thus experienced stress, experienced pain, experienced loss, swelling the cemeteries—enough to become disenchanted with all fabricated things, enough to become dispassionate, enough to be released.
SN 15.3 (translated by Ven. Thanissaro)

"The [precise working out of the] results of kamma is an unconjecturable that is not to be conjectured about, that would bring madness & vexation to anyone who conjectured about it.

"Conjecture about [the origin, etc., of] the world is an unconjecturable that is not to be conjectured about, that would bring madness & vexation to anyone who conjectured about it.
AN 4.77 (translated by Ven. Thanissaro)